A090634 -id:A090634 - cp's OEIS frontend

A090633 Start with the sequence [1, 1/2, 1/3, ..., 1/n]; form new sequence of n-1 terms by taking averages of successive terms; repeat until reach a single number F(n); a(n) = numerator of F(n).

1, 3, 7, 15, 31, 21, 127, 255, 511, 1023, 2047, 1365, 8191, 16383, 32767, 65535, 131071, 29127, 524287, 209715, 299593, 4194303, 8388607, 5592405, 33554431, 67108863, 134217727, 268435455, 536870911, 357913941, 2147483647, 4294967295, 8589934591, 17179869183

Offset: 1

N. J. A. Sloane, Dec 13 2003

F(n) is the resistance across a single resistor of an n-dimensional hypercube made of 1 ohm resistors. - Peter J. C. Moses, May 27 2004

Also, numerators of BINOMIAL transform of sequence [1, 1/2, 1/3, 1/4, ...]. - Gary W. Adamson, Apr 26 2005

			n=3: [1, 1/2, 1/3] -> [3/4, 5/6] -> [7/12], so F(3) = 7/12. Sequence of F(n)'s begins 1, 3/4, 7/12, 15/32, 31/80, 21/64, 127/448, 255/1024, ...

Putnam Competition, 2003, Problem B2.

Alois P. Heinz, Table of n, a(n) for n = 1..3320 (first 200 terms from T. D. Noe)
F. Nedermeyer and Y. Smorodinsky, Resistance in the multidimensional cube, Quantum, Sept/October 1996, pp. 12-15 (beware file is 75Mb).

Cf. A090634 (denominators).

import Data.Ratio (numerator, (%))
a090633 n = numerator z where
   [z] = (until ((== 1) . length) avg) $ map (1 %) [1..n]
   avg xs = zipWith (\x x' -> (x + x') / 2) (tail xs) xs
-- Reinhard Zumkeller, Dec 08 2011

f := proc(L) local t1,i; t1 := []; for i from 1 to nops(L)-1 do t1 := [op(t1), (L[i]+L[i+1])/2]; od: t1; end; f2 := n->[seq(1/i,i=1..n)];
F := proc(n) local L,i; L := f2(n); for i from 1 to n-1 do L := f(L); od: op(L); end;

a[n_]:=(2-2^(1-n))/n; a[1]:=1; Table[Numerator[a[n]], {n, 40}]
a[n_]:=a[n-1]+(2^(1-n)*(1+n)-2)/((n-1)*n); a[1]:=1; Table[Numerator[a[n]], {n, 40}]
a[n_]:=a[n-1]*(2^n-1)*(n-1)/(n*(2^n-2)); a[1]:=1; Table[Numerator[a[n]], {n, 40}]

From Peter J. C. Moses, May 27 2004: (Start)
F(n) = (2-2^(1-n))/n.
G.f. for F: 2*(log(1-x/2)-log(1-x)).
E.g.f. for F: Integral 2*(e^x-e^(x/2))/x dx. (End)

A212045 Numerators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.

Original entry on oeis.org

1, 3, 1, 7, 3, 5, 15, 7, 61, 2, 31, 15, 241, 25, 8, 21, 31, 131, 101, 137, 13, 127, 21, 12, 7, 2381, 343, 151, 255, 127, 2105, 167, 10781, 2033, 32663, 32, 511, 255, 16531, 929, 42061, 9383, 84677, 2357, 83, 1023, 511, 5231, 7387, 74189, 1771, 12419

Offset: 1

Peter J. C. Moses, Apr 28 2012

The term "resistance distance" for electric circuits was in use years before it was proved to be a metric (on edges of graphs). The historical meaning has been described thus: "one imagines unit resistors on each edge of a graph G and takes the resistance distance between vertices i and j of G to be the effective resistance between vertices i and j..."  (from Klein, 2002; see the References).  Let R(k,n) denote the resistance distance for k resistors in an n-dimensional cube (for details, see Example and References).  Then
R(k,n)=A212045(k,n)/A212046(k,n).  Moreover,
A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),
A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).

			First six rows of A212045/A212046:
1
3/4 .... 1
7/12 ... 3/4 .... 5/6
15/32 .. 7/12 ... 61/96 ... 2/3
31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15
21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30
The resistance distances for n=3 (the ordinary cube) are 7/12, 3/4, and 5/6, so that row 3 of the triangle of numerators is (7, 3, 5).  For the corresponding electric circuit, suppose X is a vertex of the cube. The resistance across any one of the 3 edges from X is 7/12 ohm; the resistance across any two adjoined edges (i.e., a diagonal of a face of the cubes) is 3/4 ohm; the resistance across and three adjoined edges (a diagonal of the cube) is 5/6 ohm.

F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 81-95.
D. J. Klein, Resistance-Distance Sum Rules, Croatia Chemica Acta, 75 (2002), 633-649.
Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757v1 [math.CO], 2009.
N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331-346.
David Randall, The Resistor Cube Problem.
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.

Cf. A212046, A046878, A046879, A046825, A090634, A090633.

R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;
R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)
t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]
Flatten[Numerator[t]]    (* A212045 *)
Flatten[Denominator[t]]  (* A212046 *)
TableForm[Numerator[t]]
TableForm[Denominator[t]]

A212045(n)/A212046(n) is the rational number R(k, n) =

[(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.

A212046 Denominators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.

Original entry on oeis.org

1, 4, 1, 12, 4, 6, 32, 12, 96, 3, 80, 32, 480, 48, 15, 64, 80, 320, 240, 320, 30, 448, 64, 35, 20, 6720, 960, 420, 1024, 448, 7168, 560, 35840, 6720, 107520, 105, 2304, 1024, 64512, 3584, 161280, 35840, 322560, 8960, 315, 5120, 2304, 23040, 32256

Offset: 1

Peter J. C. Moses, Apr 30 2012

The term "resistance distance" for electric circuits was in use years before it was proved to be a metric (on edges of graphs). The historical meaning has been described thus: "one imagines unit resistors on each edge of a graph G and takes the resistance distance between vertices i and j of G to be the effective resistance between vertices i and j..."  (from Klein, 2002; see the References). Let R(k,n) denote the resistance distance for k resistors in an n-dimensional cube (for details, see Example and References). Then
R(k,n)=A212045(k,n)/A212046(k,n).  Moreover,
A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),
A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).

			First six rows of A212045/A212046:
  1
  3/4 .... 1
  7/12 ... 3/4 .... 5/6
  15/32 .. 7/12 ... 61/96 ... 2/3
  31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15
  21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30
The resistance distances for n=3 (the ordinary cube) are 7/12, 3/4, and 5/6, so that row 3 of the triangle of numerators is (7, 3, 5).  For the corresponding electric circuit, suppose X is a vertex of the cube.  The resistance across any one of the 3 edges from X is 7/12 ohm; the resistance across any two adjoined edges (i.e., a diagonal of a face of the cubes) is 3/4 ohm; the resistance across and three adjoined edges (a diagonal of the cube) is 5/6 ohm.

F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 81-95.
D. J. Klein, Resistance-Distance Sum Rules, Croatia Chemica Acta, Vol. 75, No. 2 (2002), 633-649.
Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757 [math.CO], 2009.
N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331-346.
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
Wikipedia, Resistance distance

Cf. A212045, A046878, A046879, A046825, A090634, A090633.

R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;
R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)
t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]
Flatten[Numerator[t]]    (* A212045 *)
Flatten[Denominator[t]]  (* A212046 *)
TableForm[Numerator[t]]
TableForm[Denominator[t]]

A212045(n)/A212046(n) is the rational number R(k, n) =

[(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.

cp's OEIS Frontend

A090633 Start with the sequence [1, 1/2, 1/3, ..., 1/n]; form new sequence of n-1 terms by taking averages of successive terms; repeat until reach a single number F(n); a(n) = numerator of F(n).

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A212045 Numerators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.

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A212046 Denominators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.

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A131135 Denominator of (ordinary) expansion of log((x/2-1)/(x-1)).

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